Question

If a circle passes through the point (a, b) and cuts the circle x² + y² = 4 orthogonally, then the locus of its centre is ______.

Options

• 2ax – 2by – (a² + b² + 4) = 0

• 2ax + 2by – (a² + b² + 4) = 0

• 2ax – 2by + (a² + b² + 4) = 0

• 2ax + 2by + (a² + b² + 4) = 0

Answer 1

If a circle passes through the point (a, b) and cuts the circle x² + y² = 4 orthogonally, then the locus of its centre is `underlinebb(2ax + 2by - (a^2 + b^2 + 4) = 0)`.

Explanation:

Let the equation of circle is

x² + y² + 2gx + 2fy + c = 0  ......(i)

It passes through (a, b)

∴ a² + b² + 2ga + 2fb + c = 0  ......(ii)

Circle (i) cuts x² + y² = 4 orthogonally

Two circles intersect orthogonally if 2g₁g₂ + 2f₁f₂ = c₁ + c₂

∴  2(g × 0 + f × 0) = c – 4 ⇒ c = 4

∴ From (ii) a² + b² + 2ga + 2fb + 4 = 0

∴ Locus of centre (–g, –f) is a² + b² – 2ax – 2by + 4 = 0 or 2ax + 2by = a² + b² + 4